Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,8,Mod(1,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 441.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(137.761796238\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 102x^{2} + 240x + 900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 7) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 102x^{2} + 240x + 900 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 72\nu + 165 ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} + 26\nu^{2} - 288\nu - 840 ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 31\nu^{2} + 492\nu - 1815 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 5\beta _1 + 10 ) / 28 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 4\beta _1 + 100 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 36\beta_{3} - 29\beta_{2} + 362\beta _1 - 1250 ) / 14 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−19.0193 | 0 | 233.735 | 350.005 | 0 | 0 | −2011.00 | 0 | −6656.85 | ||||||||||||||||||||||||||||||
| 1.2 | −3.18552 | 0 | −117.852 | −0.588975 | 0 | 0 | 783.169 | 0 | 1.87619 | |||||||||||||||||||||||||||||||
| 1.3 | 7.38077 | 0 | −73.5243 | −290.819 | 0 | 0 | −1487.40 | 0 | −2146.47 | |||||||||||||||||||||||||||||||
| 1.4 | 20.8241 | 0 | 305.642 | 193.403 | 0 | 0 | 3699.23 | 0 | 4027.44 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 441.8.a.t | 4 | |
| 3.b | odd | 2 | 1 | 49.8.a.e | 4 | ||
| 7.b | odd | 2 | 1 | 441.8.a.s | 4 | ||
| 7.d | odd | 6 | 2 | 63.8.e.b | 8 | ||
| 21.c | even | 2 | 1 | 49.8.a.f | 4 | ||
| 21.g | even | 6 | 2 | 7.8.c.a | ✓ | 8 | |
| 21.h | odd | 6 | 2 | 49.8.c.g | 8 | ||
| 84.j | odd | 6 | 2 | 112.8.i.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.8.c.a | ✓ | 8 | 21.g | even | 6 | 2 | |
| 49.8.a.e | 4 | 3.b | odd | 2 | 1 | ||
| 49.8.a.f | 4 | 21.c | even | 2 | 1 | ||
| 49.8.c.g | 8 | 21.h | odd | 6 | 2 | ||
| 63.8.e.b | 8 | 7.d | odd | 6 | 2 | ||
| 112.8.i.c | 8 | 84.j | odd | 6 | 2 | ||
| 441.8.a.s | 4 | 7.b | odd | 2 | 1 | ||
| 441.8.a.t | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(441))\):
|
\( T_{2}^{4} - 6T_{2}^{3} - 412T_{2}^{2} + 1704T_{2} + 9312 \)
|
|
\( T_{5}^{4} - 252T_{5}^{3} - 90490T_{5}^{2} + 19632900T_{5} + 11594625 \)
|
|
\( T_{13}^{4} - 1176T_{13}^{3} - 99442952T_{13}^{2} - 32623185504T_{13} + 1618193586130576 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 6 T^{3} + \cdots + 9312 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 252 T^{3} + \cdots + 11594625 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 7298108160225 \)
|
| $13$ |
\( T^{4} + \cdots + 16\!\cdots\!76 \)
|
| $17$ |
\( T^{4} + \cdots - 11\!\cdots\!27 \)
|
| $19$ |
\( T^{4} + \cdots - 52\!\cdots\!47 \)
|
| $23$ |
\( T^{4} + \cdots - 77\!\cdots\!63 \)
|
| $29$ |
\( T^{4} + \cdots - 59\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 43\!\cdots\!31 \)
|
| $37$ |
\( T^{4} + \cdots + 38\!\cdots\!93 \)
|
| $41$ |
\( T^{4} + \cdots - 57\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 32\!\cdots\!56 \)
|
| $47$ |
\( T^{4} + \cdots - 33\!\cdots\!75 \)
|
| $53$ |
\( T^{4} + \cdots + 43\!\cdots\!13 \)
|
| $59$ |
\( T^{4} + \cdots + 16\!\cdots\!25 \)
|
| $61$ |
\( T^{4} + \cdots - 56\!\cdots\!11 \)
|
| $67$ |
\( T^{4} + \cdots + 19\!\cdots\!25 \)
|
| $71$ |
\( T^{4} + \cdots - 23\!\cdots\!24 \)
|
| $73$ |
\( T^{4} + \cdots + 89\!\cdots\!13 \)
|
| $79$ |
\( T^{4} + \cdots - 17\!\cdots\!23 \)
|
| $83$ |
\( T^{4} + \cdots - 62\!\cdots\!16 \)
|
| $89$ |
\( T^{4} + \cdots - 24\!\cdots\!87 \)
|
| $97$ |
\( T^{4} + \cdots + 42\!\cdots\!36 \)
|
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